Theorem int0 4926

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4458	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4918	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 234	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 267	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2759	1 ⊢ ∩ ∅ = V

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2142  ∀wral 3078  Vcvv 3454  ∅c0 4285  ∩ cint 4911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3456  df-dif 3907  df-nul 4286  df-int 4912

This theorem is used by:  unissint  4936  uniintsn  4949  rint0  4952  intex  5313  intnex  5314  oev2  8506  fiint  9284  elfi2  9372  fi0  9378  cardmin2  9992  00lsp  21113  cmpfi  23576  ptbasfi  23749  fbssint  24006  fclscmp  24198  zarcmplem  34280  rankeq1o  36671  bj-0int  37771  heibor1lem  38488  ipoglb0  49800  mreclat  49803